Almost-invariant and essentially-invariant halfspaces.

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if X is an infinite-dimensional complex Banach space then every operator T ∈ L ( X ) admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. [ABSTRACT FROM AUTHOR]